remap MCMC output via affine transformations

Description

Remap the MCMC iterates in an ideal object via an affine transformation, imposing identifying restrictions ex post (aka post-processing).

Usage

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postProcess(object, constraints="normalize", debug = FALSE)

Arguments

object

an object of class ideal

constraints

list of length d+1, each component providing a set of d restrictions, where d is the dimension of the fitted ideal model; or the character string normalize (default). If a list, the name of each component should uniquely match a legislator/subject's name. See Details.

debug

logical flag for verbose output, used for debugging

Details

Item-response models are unidentified without restrictions on the underlying parameters. Consider the d=1 dimensional case. The model is

Pr(y_[ij] = 1) = F(x_i b_j - a_j).

Any linear transformation of the latent traits, say,

x* = mx + c

can be exactly offset by applying the appropriate linear transformations to the item/bill parameters, meaning that there is no unique set of values for the model parameters that will maximize the likelihood function. In higher dimensions, the latent traits can also be transformed via any arbitrary rotation, dilation and translation, with offsetting transformations applied to the item/bill parameters.

One strategy in MCMC is to ignore the lack of identification at run time, but apply identifying restrictions ex post, “post-processing” the MCMC output, iteration-by-iteration. In a d-dimensional IRT model, a sufficient condition for global identification is to fix d+1 latent traits, provided the constrained latent traits span the d dimensional latent space. This function implements this strategy. The user supplies a set of constrained ideal points in the constraints list. The function then processes the MCMC output in the ideal object, finding the transformation that maps the current iteration's sampled values for x (latent traits/ideal points) into the sub-space of identified parameters defined by the fixed points in constraints; i.e., what is the affine transformation that maps the unconstrained ideal points into the constraints? Aside from miniscule numerical inaccuracies resulting from matrix inversion etc, this transformation is exact: after post-processing, the d+1 constrained points do not vary over the MCMC iterations. The remaining n-d-1 ideal points are subject to (posterior) uncertainty; the “random tour” of the joint parameter space of these parameters produced by the MCMC algorithm has been mapped into a subspace in which the parameters are globally identified.

If the ideal object was produced with store.item set to TRUE, then the item parameters are also post-processed, applying the inverse transformation. Specifically, recall that the IRT model is

Pr(y_[ij] = 1) = F(x_i' b_j)

where in this formulation x_i is a vector of length d+1, including a -1 to put a constant term into the model (i.e., the intercept or difficulty parameter is part of beta_j). Let A denote the non-singular, d+1-by-d+1 matrix that maps the x into the space of identified parameters. Recall that this transformation is computed iteration by iteration. Then each x_i is transformed to x*_i = Ax_i and b_j is transformed to b_j^* = A^(-1) b_j, i = 1, …, n; j = 1, …, m.

Local identification can be obtained for a one-dimensional model by simply imposing a normalizing restriction on the ideal points: this normalization (mean zero, standard deviation one) is the default behavior, but (a) is only sufficient for local identification when the rollcall object was fit with d=1; (b) is not sufficient for even local identification when d>1, with further restrictions required so as to rule out other forms of invariance (e.g., translation, or "dimension-switching", a phenomenon akin to label-switching in mixture modeling).

The default is to impose dimension-by-dimension normalization with respect to the means of the marginal posterior densities of the ideal points, such that the these means (the usual Bayes estimates of the ideal points) have mean zero and standard deviation one across legislators. An offsetting transformation is applied to the items parameters as well, if they are saved in the ideal object.

Specifically, in one-dimension, the two-parameter IRT model is

Pr(y_[ij] = 1) = F(x_i b_j - a_j).

If we normalize the x_i to x*_i = (x_i - c)/m then the offsetting transformations for the item/bill parameters are b*_j = b_j m and a*_j = a_j - cb_j.

Value

An object of class ideal, with components suitably transformed and recomputed (i.e., x is transformed and xbar recomputed, and if the ideal object was fit with store.item=TRUE, beta is transformed and betabar is recomputed).

Note

Applying transformations to obtain identification can sometimes lead to surprising results. Each data point makes the same likelihood contributions with either the identified or unidentified parameters. But, in general, predictions generated with the parameters set to their posterior means will differ depending on whether one uses the identified subset of parameters or the unidentified parameters. For this reason, caution should be used when using a function such as predict after post-processing output from ideal. A better strategy is to compute the estimand of interest at each iteration and then take averages over iterations.

When specifying a value of burnin different from that used in fitting the ideal object, note a distinction between the iteration numbers of the stored iterations, and the number of stored iterations. That is, the n-th iteration stored in an ideal object will not be iteration n if the user specified thin>1 in the call to ideal. Here, iterations are tagged with their iteration number. Thus, if the user called ideal with thin=10 and burnin=100 then the stored iterations are numbered 100, 110, 120, .... Any future subsetting via a burnin refers to this iteration number.

Author(s)

Simon Jackman jackman@stanford.edu

References

Hoff, Peter, Adrian E. Raftery and Mark S. Handcock. 2002. Latent Space Approaches to Social Network Analysis. Journal of the American Statistical Association 97:1090–1098.

Edwards, Yancy D. and Greg M. Allenby. 2003. Multivariate Analysis of Mulitple Response Data. Journal of Marketing Research 40:321–334.

Rivers, Douglas. 2003. “Identification of Multidimensional Item-Response Models.” Typescript. Department of Political Science, Stanford University.

Examples

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data(s109)
f = system.file("extdata",package="pscl","id1.rda")
load(f)

id1Local <- postProcess(id1)    ## default is to normalize
summary(id1Local)

id1pp <- postProcess(id1,
                     constraints=list(BOXER=-1,INHOFE=1))
summary(id1pp)

## two-dimensional fit
f = system.file("extdata",package="pscl","id2.rda")
load(f)

id2pp <- postProcess(id2,
                     constraints=list(BOXER=c(-1,0),
                       INHOFE=c(1,0),
                       CHAFEE=c(0,.25)))

tracex(id2pp,d=1:2,
       legis=c("BOXER","INHOFE","COLLINS","FEINGOLD","COLEMAN",
         "CHAFEE","MCCAIN","KYL"))

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